Properties

Label 507.4.b.e.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-17})\)
Defining polynomial: \( x^{4} - 17x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(3.57071 - 2.06155i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.e.337.4

$q$-expansion

\(f(q)\) \(=\) \(q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +3.05006i q^{5} +12.3693i q^{6} -6.68324i q^{7} +4.12311i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +3.05006i q^{5} +12.3693i q^{6} -6.68324i q^{7} +4.12311i q^{8} +9.00000 q^{9} +12.5757 q^{10} +32.2500i q^{11} +27.0000 q^{12} -27.5557 q^{14} -9.15018i q^{15} -55.0000 q^{16} +28.8586 q^{17} -37.1080i q^{18} +101.194i q^{19} -27.4505i q^{20} +20.0497i q^{21} +132.970 q^{22} +118.990 q^{23} -12.3693i q^{24} +115.697 q^{25} -27.0000 q^{27} +60.1492i q^{28} +160.111 q^{29} -37.7271 q^{30} +38.0705i q^{31} +259.756i q^{32} -96.7499i q^{33} -118.987i q^{34} +20.3843 q^{35} -81.0000 q^{36} -327.568i q^{37} +417.233 q^{38} -12.5757 q^{40} -56.0846i q^{41} +82.6671 q^{42} -127.879 q^{43} -290.250i q^{44} +27.4505i q^{45} -490.608i q^{46} -517.983i q^{47} +165.000 q^{48} +298.334 q^{49} -477.032i q^{50} -86.5757 q^{51} -695.546 q^{53} +111.324i q^{54} -98.3643 q^{55} +27.5557 q^{56} -303.581i q^{57} -660.156i q^{58} +656.523i q^{59} +82.3516i q^{60} +701.304 q^{61} +156.969 q^{62} -60.1492i q^{63} +631.000 q^{64} -398.910 q^{66} +57.1750i q^{67} -259.727 q^{68} -356.970 q^{69} -84.0465i q^{70} -309.226i q^{71} +37.1080i q^{72} -389.711i q^{73} -1350.60 q^{74} -347.091 q^{75} -910.744i q^{76} +215.534 q^{77} +901.820 q^{79} -167.753i q^{80} +81.0000 q^{81} -231.243 q^{82} -687.095i q^{83} -180.448i q^{84} +88.0203i q^{85} +527.257i q^{86} -480.334 q^{87} -132.970 q^{88} +1070.54i q^{89} +113.181 q^{90} -1070.91 q^{92} -114.211i q^{93} -2135.70 q^{94} -308.647 q^{95} -779.267i q^{96} -1754.48i q^{97} -1230.06i q^{98} +290.250i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.12311i − 1.45774i −0.684653 0.728869i \(-0.740046\pi\)
0.684653 0.728869i \(-0.259954\pi\)
\(3\) −3.00000 −0.577350
\(4\) −9.00000 −1.12500
\(5\) 3.05006i 0.272806i 0.990653 + 0.136403i \(0.0435541\pi\)
−0.990653 + 0.136403i \(0.956446\pi\)
\(6\) 12.3693i 0.841625i
\(7\) − 6.68324i − 0.360861i −0.983588 0.180431i \(-0.942251\pi\)
0.983588 0.180431i \(-0.0577491\pi\)
\(8\) 4.12311i 0.182217i
\(9\) 9.00000 0.333333
\(10\) 12.5757 0.397679
\(11\) 32.2500i 0.883975i 0.897021 + 0.441988i \(0.145727\pi\)
−0.897021 + 0.441988i \(0.854273\pi\)
\(12\) 27.0000 0.649519
\(13\) 0 0
\(14\) −27.5557 −0.526041
\(15\) − 9.15018i − 0.157504i
\(16\) −55.0000 −0.859375
\(17\) 28.8586 0.411720 0.205860 0.978582i \(-0.434001\pi\)
0.205860 + 0.978582i \(0.434001\pi\)
\(18\) − 37.1080i − 0.485913i
\(19\) 101.194i 1.22187i 0.791682 + 0.610933i \(0.209206\pi\)
−0.791682 + 0.610933i \(0.790794\pi\)
\(20\) − 27.4505i − 0.306906i
\(21\) 20.0497i 0.208343i
\(22\) 132.970 1.28860
\(23\) 118.990 1.07874 0.539372 0.842067i \(-0.318662\pi\)
0.539372 + 0.842067i \(0.318662\pi\)
\(24\) − 12.3693i − 0.105203i
\(25\) 115.697 0.925577
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 60.1492i 0.405969i
\(29\) 160.111 1.02524 0.512620 0.858616i \(-0.328675\pi\)
0.512620 + 0.858616i \(0.328675\pi\)
\(30\) −37.7271 −0.229600
\(31\) 38.0705i 0.220570i 0.993900 + 0.110285i \(0.0351763\pi\)
−0.993900 + 0.110285i \(0.964824\pi\)
\(32\) 259.756i 1.43496i
\(33\) − 96.7499i − 0.510363i
\(34\) − 118.987i − 0.600179i
\(35\) 20.3843 0.0984449
\(36\) −81.0000 −0.375000
\(37\) − 327.568i − 1.45545i −0.685866 0.727727i \(-0.740577\pi\)
0.685866 0.727727i \(-0.259423\pi\)
\(38\) 417.233 1.78116
\(39\) 0 0
\(40\) −12.5757 −0.0497099
\(41\) − 56.0846i − 0.213633i −0.994279 0.106816i \(-0.965934\pi\)
0.994279 0.106816i \(-0.0340657\pi\)
\(42\) 82.6671 0.303710
\(43\) −127.879 −0.453519 −0.226759 0.973951i \(-0.572813\pi\)
−0.226759 + 0.973951i \(0.572813\pi\)
\(44\) − 290.250i − 0.994472i
\(45\) 27.4505i 0.0909352i
\(46\) − 490.608i − 1.57253i
\(47\) − 517.983i − 1.60757i −0.594923 0.803783i \(-0.702817\pi\)
0.594923 0.803783i \(-0.297183\pi\)
\(48\) 165.000 0.496160
\(49\) 298.334 0.869779
\(50\) − 477.032i − 1.34925i
\(51\) −86.5757 −0.237706
\(52\) 0 0
\(53\) −695.546 −1.80265 −0.901326 0.433141i \(-0.857405\pi\)
−0.901326 + 0.433141i \(0.857405\pi\)
\(54\) 111.324i 0.280542i
\(55\) −98.3643 −0.241153
\(56\) 27.5557 0.0657551
\(57\) − 303.581i − 0.705445i
\(58\) − 660.156i − 1.49453i
\(59\) 656.523i 1.44868i 0.689444 + 0.724339i \(0.257855\pi\)
−0.689444 + 0.724339i \(0.742145\pi\)
\(60\) 82.3516i 0.177192i
\(61\) 701.304 1.47201 0.736007 0.676974i \(-0.236709\pi\)
0.736007 + 0.676974i \(0.236709\pi\)
\(62\) 156.969 0.321533
\(63\) − 60.1492i − 0.120287i
\(64\) 631.000 1.23242
\(65\) 0 0
\(66\) −398.910 −0.743976
\(67\) 57.1750i 0.104254i 0.998640 + 0.0521271i \(0.0166001\pi\)
−0.998640 + 0.0521271i \(0.983400\pi\)
\(68\) −259.727 −0.463184
\(69\) −356.970 −0.622814
\(70\) − 84.0465i − 0.143507i
\(71\) − 309.226i − 0.516879i −0.966027 0.258440i \(-0.916792\pi\)
0.966027 0.258440i \(-0.0832083\pi\)
\(72\) 37.1080i 0.0607391i
\(73\) − 389.711i − 0.624826i −0.949946 0.312413i \(-0.898863\pi\)
0.949946 0.312413i \(-0.101137\pi\)
\(74\) −1350.60 −2.12167
\(75\) −347.091 −0.534382
\(76\) − 910.744i − 1.37460i
\(77\) 215.534 0.318992
\(78\) 0 0
\(79\) 901.820 1.28434 0.642169 0.766563i \(-0.278035\pi\)
0.642169 + 0.766563i \(0.278035\pi\)
\(80\) − 167.753i − 0.234442i
\(81\) 81.0000 0.111111
\(82\) −231.243 −0.311421
\(83\) − 687.095i − 0.908657i −0.890834 0.454328i \(-0.849879\pi\)
0.890834 0.454328i \(-0.150121\pi\)
\(84\) − 180.448i − 0.234386i
\(85\) 88.0203i 0.112319i
\(86\) 527.257i 0.661111i
\(87\) −480.334 −0.591922
\(88\) −132.970 −0.161076
\(89\) 1070.54i 1.27502i 0.770442 + 0.637510i \(0.220036\pi\)
−0.770442 + 0.637510i \(0.779964\pi\)
\(90\) 113.181 0.132560
\(91\) 0 0
\(92\) −1070.91 −1.21359
\(93\) − 114.211i − 0.127346i
\(94\) −2135.70 −2.34341
\(95\) −308.647 −0.333332
\(96\) − 779.267i − 0.828475i
\(97\) − 1754.48i − 1.83650i −0.395998 0.918251i \(-0.629601\pi\)
0.395998 0.918251i \(-0.370399\pi\)
\(98\) − 1230.06i − 1.26791i
\(99\) 290.250i 0.294658i
\(100\) −1041.27 −1.04127
\(101\) 640.840 0.631346 0.315673 0.948868i \(-0.397770\pi\)
0.315673 + 0.948868i \(0.397770\pi\)
\(102\) 356.961i 0.346514i
\(103\) 693.153 0.663091 0.331546 0.943439i \(-0.392430\pi\)
0.331546 + 0.943439i \(0.392430\pi\)
\(104\) 0 0
\(105\) −61.1528 −0.0568372
\(106\) 2867.81i 2.62779i
\(107\) −405.676 −0.366525 −0.183262 0.983064i \(-0.558666\pi\)
−0.183262 + 0.983064i \(0.558666\pi\)
\(108\) 243.000 0.216506
\(109\) 479.516i 0.421370i 0.977554 + 0.210685i \(0.0675694\pi\)
−0.977554 + 0.210685i \(0.932431\pi\)
\(110\) 405.566i 0.351538i
\(111\) 982.704i 0.840307i
\(112\) 367.578i 0.310115i
\(113\) −1547.20 −1.28804 −0.644021 0.765008i \(-0.722735\pi\)
−0.644021 + 0.765008i \(0.722735\pi\)
\(114\) −1251.70 −1.02835
\(115\) 362.926i 0.294288i
\(116\) −1441.00 −1.15339
\(117\) 0 0
\(118\) 2706.91 2.11179
\(119\) − 192.869i − 0.148574i
\(120\) 37.7271 0.0287000
\(121\) 290.940 0.218588
\(122\) − 2891.55i − 2.14581i
\(123\) 168.254i 0.123341i
\(124\) − 342.634i − 0.248141i
\(125\) 734.140i 0.525308i
\(126\) −248.001 −0.175347
\(127\) 2495.15 1.74338 0.871690 0.490059i \(-0.163025\pi\)
0.871690 + 0.490059i \(0.163025\pi\)
\(128\) − 523.634i − 0.361587i
\(129\) 383.636 0.261839
\(130\) 0 0
\(131\) 43.7571 0.0291838 0.0145919 0.999894i \(-0.495355\pi\)
0.0145919 + 0.999894i \(0.495355\pi\)
\(132\) 870.749i 0.574159i
\(133\) 676.303 0.440924
\(134\) 235.739 0.151975
\(135\) − 82.3516i − 0.0525015i
\(136\) 118.987i 0.0750224i
\(137\) − 206.194i − 0.128586i −0.997931 0.0642932i \(-0.979521\pi\)
0.997931 0.0642932i \(-0.0204793\pi\)
\(138\) 1471.83i 0.907899i
\(139\) −100.000 −0.0610208 −0.0305104 0.999534i \(-0.509713\pi\)
−0.0305104 + 0.999534i \(0.509713\pi\)
\(140\) −183.459 −0.110751
\(141\) 1553.95i 0.928128i
\(142\) −1274.97 −0.753474
\(143\) 0 0
\(144\) −495.000 −0.286458
\(145\) 488.349i 0.279691i
\(146\) −1606.82 −0.910832
\(147\) −895.003 −0.502167
\(148\) 2948.11i 1.63739i
\(149\) − 380.451i − 0.209179i −0.994515 0.104590i \(-0.966647\pi\)
0.994515 0.104590i \(-0.0333529\pi\)
\(150\) 1431.09i 0.778989i
\(151\) − 1517.45i − 0.817805i −0.912578 0.408902i \(-0.865912\pi\)
0.912578 0.408902i \(-0.134088\pi\)
\(152\) −417.233 −0.222645
\(153\) 259.727 0.137240
\(154\) − 888.671i − 0.465007i
\(155\) −116.117 −0.0601726
\(156\) 0 0
\(157\) 1450.16 0.737166 0.368583 0.929595i \(-0.379843\pi\)
0.368583 + 0.929595i \(0.379843\pi\)
\(158\) − 3718.30i − 1.87223i
\(159\) 2086.64 1.04076
\(160\) −792.270 −0.391465
\(161\) − 795.239i − 0.389277i
\(162\) − 333.972i − 0.161971i
\(163\) 2342.36i 1.12557i 0.826603 + 0.562785i \(0.190270\pi\)
−0.826603 + 0.562785i \(0.809730\pi\)
\(164\) 504.762i 0.240337i
\(165\) 295.093 0.139230
\(166\) −2832.97 −1.32458
\(167\) − 40.0731i − 0.0185686i −0.999957 0.00928428i \(-0.997045\pi\)
0.999957 0.00928428i \(-0.00295532\pi\)
\(168\) −82.6671 −0.0379637
\(169\) 0 0
\(170\) 362.917 0.163732
\(171\) 910.744i 0.407289i
\(172\) 1150.91 0.510208
\(173\) −1909.54 −0.839189 −0.419594 0.907712i \(-0.637828\pi\)
−0.419594 + 0.907712i \(0.637828\pi\)
\(174\) 1980.47i 0.862868i
\(175\) − 773.232i − 0.334005i
\(176\) − 1773.75i − 0.759666i
\(177\) − 1969.57i − 0.836395i
\(178\) 4413.94 1.85864
\(179\) 509.959 0.212939 0.106470 0.994316i \(-0.466045\pi\)
0.106470 + 0.994316i \(0.466045\pi\)
\(180\) − 247.055i − 0.102302i
\(181\) 2136.88 0.877531 0.438766 0.898602i \(-0.355416\pi\)
0.438766 + 0.898602i \(0.355416\pi\)
\(182\) 0 0
\(183\) −2103.91 −0.849867
\(184\) 490.608i 0.196566i
\(185\) 999.101 0.397056
\(186\) −470.906 −0.185637
\(187\) 930.688i 0.363950i
\(188\) 4661.85i 1.80851i
\(189\) 180.448i 0.0694478i
\(190\) 1272.58i 0.485911i
\(191\) 4057.74 1.53721 0.768607 0.639721i \(-0.220950\pi\)
0.768607 + 0.639721i \(0.220950\pi\)
\(192\) −1893.00 −0.711539
\(193\) 873.394i 0.325742i 0.986647 + 0.162871i \(0.0520755\pi\)
−0.986647 + 0.162871i \(0.947925\pi\)
\(194\) −7233.92 −2.67714
\(195\) 0 0
\(196\) −2685.01 −0.978502
\(197\) − 4147.25i − 1.49989i −0.661498 0.749947i \(-0.730079\pi\)
0.661498 0.749947i \(-0.269921\pi\)
\(198\) 1196.73 0.429535
\(199\) 2404.06 0.856379 0.428189 0.903689i \(-0.359152\pi\)
0.428189 + 0.903689i \(0.359152\pi\)
\(200\) 477.032i 0.168656i
\(201\) − 171.525i − 0.0601912i
\(202\) − 2642.25i − 0.920337i
\(203\) − 1070.06i − 0.369969i
\(204\) 779.181 0.267420
\(205\) 171.061 0.0582802
\(206\) − 2857.94i − 0.966613i
\(207\) 1070.91 0.359582
\(208\) 0 0
\(209\) −3263.50 −1.08010
\(210\) 252.140i 0.0828538i
\(211\) 3868.85 1.26229 0.631144 0.775665i \(-0.282586\pi\)
0.631144 + 0.775665i \(0.282586\pi\)
\(212\) 6259.91 2.02798
\(213\) 927.679i 0.298420i
\(214\) 1672.64i 0.534297i
\(215\) − 390.037i − 0.123722i
\(216\) − 111.324i − 0.0350677i
\(217\) 254.434 0.0795950
\(218\) 1977.09 0.614246
\(219\) 1169.13i 0.360743i
\(220\) 885.279 0.271298
\(221\) 0 0
\(222\) 4051.79 1.22495
\(223\) − 2813.55i − 0.844885i −0.906390 0.422443i \(-0.861173\pi\)
0.906390 0.422443i \(-0.138827\pi\)
\(224\) 1736.01 0.517822
\(225\) 1041.27 0.308526
\(226\) 6379.29i 1.87763i
\(227\) 4518.37i 1.32112i 0.750772 + 0.660561i \(0.229682\pi\)
−0.750772 + 0.660561i \(0.770318\pi\)
\(228\) 2732.23i 0.793625i
\(229\) − 1305.27i − 0.376658i −0.982106 0.188329i \(-0.939693\pi\)
0.982106 0.188329i \(-0.0603070\pi\)
\(230\) 1496.38 0.428994
\(231\) −646.603 −0.184170
\(232\) 660.156i 0.186816i
\(233\) 3360.55 0.944879 0.472440 0.881363i \(-0.343373\pi\)
0.472440 + 0.881363i \(0.343373\pi\)
\(234\) 0 0
\(235\) 1579.88 0.438553
\(236\) − 5908.71i − 1.62976i
\(237\) −2705.46 −0.741513
\(238\) −795.219 −0.216581
\(239\) 4737.17i 1.28210i 0.767499 + 0.641050i \(0.221501\pi\)
−0.767499 + 0.641050i \(0.778499\pi\)
\(240\) 503.260i 0.135355i
\(241\) 4785.28i 1.27903i 0.768778 + 0.639516i \(0.220865\pi\)
−0.768778 + 0.639516i \(0.779135\pi\)
\(242\) − 1199.58i − 0.318643i
\(243\) −243.000 −0.0641500
\(244\) −6311.74 −1.65601
\(245\) 909.937i 0.237281i
\(246\) 693.729 0.179799
\(247\) 0 0
\(248\) −156.969 −0.0401916
\(249\) 2061.29i 0.524613i
\(250\) 3026.94 0.765762
\(251\) 3273.86 0.823284 0.411642 0.911346i \(-0.364955\pi\)
0.411642 + 0.911346i \(0.364955\pi\)
\(252\) 541.343i 0.135323i
\(253\) 3837.42i 0.953584i
\(254\) − 10287.8i − 2.54139i
\(255\) − 264.061i − 0.0648476i
\(256\) 2889.00 0.705322
\(257\) 6545.81 1.58878 0.794390 0.607408i \(-0.207791\pi\)
0.794390 + 0.607408i \(0.207791\pi\)
\(258\) − 1581.77i − 0.381693i
\(259\) −2189.22 −0.525217
\(260\) 0 0
\(261\) 1441.00 0.341746
\(262\) − 180.415i − 0.0425424i
\(263\) 88.2014 0.0206796 0.0103398 0.999947i \(-0.496709\pi\)
0.0103398 + 0.999947i \(0.496709\pi\)
\(264\) 398.910 0.0929970
\(265\) − 2121.46i − 0.491773i
\(266\) − 2788.47i − 0.642752i
\(267\) − 3211.61i − 0.736133i
\(268\) − 514.575i − 0.117286i
\(269\) −4527.60 −1.02622 −0.513109 0.858324i \(-0.671506\pi\)
−0.513109 + 0.858324i \(0.671506\pi\)
\(270\) −339.544 −0.0765334
\(271\) 8321.82i 1.86537i 0.360695 + 0.932684i \(0.382540\pi\)
−0.360695 + 0.932684i \(0.617460\pi\)
\(272\) −1587.22 −0.353821
\(273\) 0 0
\(274\) −850.160 −0.187445
\(275\) 3731.23i 0.818187i
\(276\) 3212.73 0.700665
\(277\) 2881.31 0.624986 0.312493 0.949920i \(-0.398836\pi\)
0.312493 + 0.949920i \(0.398836\pi\)
\(278\) 412.311i 0.0889523i
\(279\) 342.634i 0.0735232i
\(280\) 84.0465i 0.0179384i
\(281\) − 2817.99i − 0.598247i −0.954214 0.299123i \(-0.903306\pi\)
0.954214 0.299123i \(-0.0966942\pi\)
\(282\) 6407.10 1.35297
\(283\) −264.601 −0.0555792 −0.0277896 0.999614i \(-0.508847\pi\)
−0.0277896 + 0.999614i \(0.508847\pi\)
\(284\) 2783.04i 0.581489i
\(285\) 925.941 0.192449
\(286\) 0 0
\(287\) −374.827 −0.0770918
\(288\) 2337.80i 0.478320i
\(289\) −4080.18 −0.830487
\(290\) 2013.52 0.407716
\(291\) 5263.45i 1.06031i
\(292\) 3507.40i 0.702929i
\(293\) 4292.52i 0.855877i 0.903808 + 0.427938i \(0.140760\pi\)
−0.903808 + 0.427938i \(0.859240\pi\)
\(294\) 3690.19i 0.732028i
\(295\) −2002.43 −0.395208
\(296\) 1350.60 0.265209
\(297\) − 870.749i − 0.170121i
\(298\) −1568.64 −0.304929
\(299\) 0 0
\(300\) 3123.82 0.601180
\(301\) 854.643i 0.163657i
\(302\) −6256.62 −1.19214
\(303\) −1922.52 −0.364508
\(304\) − 5565.66i − 1.05004i
\(305\) 2139.02i 0.401573i
\(306\) − 1070.88i − 0.200060i
\(307\) 7026.26i 1.30622i 0.757263 + 0.653110i \(0.226536\pi\)
−0.757263 + 0.653110i \(0.773464\pi\)
\(308\) −1939.81 −0.358866
\(309\) −2079.46 −0.382836
\(310\) 478.763i 0.0877159i
\(311\) 1133.21 0.206618 0.103309 0.994649i \(-0.467057\pi\)
0.103309 + 0.994649i \(0.467057\pi\)
\(312\) 0 0
\(313\) −5285.95 −0.954566 −0.477283 0.878750i \(-0.658378\pi\)
−0.477283 + 0.878750i \(0.658378\pi\)
\(314\) − 5979.15i − 1.07459i
\(315\) 183.459 0.0328150
\(316\) −8116.38 −1.44488
\(317\) 4782.16i 0.847296i 0.905827 + 0.423648i \(0.139251\pi\)
−0.905827 + 0.423648i \(0.860749\pi\)
\(318\) − 8603.43i − 1.51716i
\(319\) 5163.59i 0.906286i
\(320\) 1924.59i 0.336212i
\(321\) 1217.03 0.211613
\(322\) −3278.85 −0.567464
\(323\) 2920.31i 0.503066i
\(324\) −729.000 −0.125000
\(325\) 0 0
\(326\) 9657.80 1.64079
\(327\) − 1438.55i − 0.243278i
\(328\) 231.243 0.0389276
\(329\) −3461.81 −0.580108
\(330\) − 1216.70i − 0.202961i
\(331\) − 8669.98i − 1.43971i −0.694122 0.719857i \(-0.744207\pi\)
0.694122 0.719857i \(-0.255793\pi\)
\(332\) 6183.86i 1.02224i
\(333\) − 2948.11i − 0.485152i
\(334\) −165.226 −0.0270681
\(335\) −174.387 −0.0284412
\(336\) − 1102.73i − 0.179045i
\(337\) −8526.59 −1.37826 −0.689129 0.724639i \(-0.742007\pi\)
−0.689129 + 0.724639i \(0.742007\pi\)
\(338\) 0 0
\(339\) 4641.61 0.743651
\(340\) − 792.183i − 0.126359i
\(341\) −1227.77 −0.194978
\(342\) 3755.10 0.593720
\(343\) − 4286.19i − 0.674731i
\(344\) − 527.257i − 0.0826389i
\(345\) − 1088.78i − 0.169907i
\(346\) 7873.23i 1.22332i
\(347\) −12581.5 −1.94643 −0.973213 0.229907i \(-0.926158\pi\)
−0.973213 + 0.229907i \(0.926158\pi\)
\(348\) 4323.01 0.665913
\(349\) − 8961.18i − 1.37444i −0.726447 0.687222i \(-0.758830\pi\)
0.726447 0.687222i \(-0.241170\pi\)
\(350\) −3188.12 −0.486892
\(351\) 0 0
\(352\) −8377.11 −1.26847
\(353\) 5357.99i 0.807868i 0.914788 + 0.403934i \(0.132357\pi\)
−0.914788 + 0.403934i \(0.867643\pi\)
\(354\) −8120.74 −1.21924
\(355\) 943.159 0.141007
\(356\) − 9634.84i − 1.43440i
\(357\) 578.606i 0.0857790i
\(358\) − 2102.61i − 0.310409i
\(359\) − 2705.40i − 0.397731i −0.980027 0.198866i \(-0.936274\pi\)
0.980027 0.198866i \(-0.0637257\pi\)
\(360\) −113.181 −0.0165700
\(361\) −3381.19 −0.492957
\(362\) − 8810.59i − 1.27921i
\(363\) −872.820 −0.126202
\(364\) 0 0
\(365\) 1188.64 0.170456
\(366\) 8674.65i 1.23888i
\(367\) 10473.8 1.48972 0.744858 0.667223i \(-0.232517\pi\)
0.744858 + 0.667223i \(0.232517\pi\)
\(368\) −6544.45 −0.927046
\(369\) − 504.762i − 0.0712110i
\(370\) − 4119.40i − 0.578804i
\(371\) 4648.50i 0.650507i
\(372\) 1027.90i 0.143264i
\(373\) −12763.0 −1.77170 −0.885850 0.463973i \(-0.846424\pi\)
−0.885850 + 0.463973i \(0.846424\pi\)
\(374\) 3837.32 0.530544
\(375\) − 2202.42i − 0.303287i
\(376\) 2135.70 0.292926
\(377\) 0 0
\(378\) 744.004 0.101237
\(379\) 2318.02i 0.314166i 0.987585 + 0.157083i \(0.0502090\pi\)
−0.987585 + 0.157083i \(0.949791\pi\)
\(380\) 2777.82 0.374998
\(381\) −7485.46 −1.00654
\(382\) − 16730.5i − 2.24086i
\(383\) − 1983.34i − 0.264606i −0.991209 0.132303i \(-0.957763\pi\)
0.991209 0.132303i \(-0.0422372\pi\)
\(384\) 1570.90i 0.208763i
\(385\) 657.392i 0.0870229i
\(386\) 3601.10 0.474847
\(387\) −1150.91 −0.151173
\(388\) 15790.3i 2.06607i
\(389\) 3244.51 0.422887 0.211444 0.977390i \(-0.432184\pi\)
0.211444 + 0.977390i \(0.432184\pi\)
\(390\) 0 0
\(391\) 3433.88 0.444140
\(392\) 1230.06i 0.158489i
\(393\) −131.271 −0.0168493
\(394\) −17099.5 −2.18645
\(395\) 2750.60i 0.350374i
\(396\) − 2612.25i − 0.331491i
\(397\) 3759.72i 0.475302i 0.971351 + 0.237651i \(0.0763775\pi\)
−0.971351 + 0.237651i \(0.923622\pi\)
\(398\) − 9912.20i − 1.24838i
\(399\) −2028.91 −0.254568
\(400\) −6363.34 −0.795418
\(401\) − 1997.55i − 0.248760i −0.992235 0.124380i \(-0.960306\pi\)
0.992235 0.124380i \(-0.0396941\pi\)
\(402\) −707.216 −0.0877431
\(403\) 0 0
\(404\) −5767.56 −0.710264
\(405\) 247.055i 0.0303117i
\(406\) −4411.98 −0.539318
\(407\) 10564.1 1.28659
\(408\) − 356.961i − 0.0433142i
\(409\) 5195.23i 0.628087i 0.949409 + 0.314044i \(0.101684\pi\)
−0.949409 + 0.314044i \(0.898316\pi\)
\(410\) − 705.304i − 0.0849573i
\(411\) 618.582i 0.0742394i
\(412\) −6238.38 −0.745977
\(413\) 4387.70 0.522772
\(414\) − 4415.48i − 0.524176i
\(415\) 2095.68 0.247887
\(416\) 0 0
\(417\) 300.000 0.0352304
\(418\) 13455.7i 1.57450i
\(419\) −6822.11 −0.795422 −0.397711 0.917511i \(-0.630195\pi\)
−0.397711 + 0.917511i \(0.630195\pi\)
\(420\) 550.376 0.0639419
\(421\) − 7537.70i − 0.872601i −0.899801 0.436300i \(-0.856288\pi\)
0.899801 0.436300i \(-0.143712\pi\)
\(422\) − 15951.7i − 1.84009i
\(423\) − 4661.85i − 0.535855i
\(424\) − 2867.81i − 0.328474i
\(425\) 3338.85 0.381078
\(426\) 3824.92 0.435019
\(427\) − 4686.99i − 0.531192i
\(428\) 3651.08 0.412340
\(429\) 0 0
\(430\) −1608.16 −0.180355
\(431\) 13404.2i 1.49805i 0.662544 + 0.749023i \(0.269477\pi\)
−0.662544 + 0.749023i \(0.730523\pi\)
\(432\) 1485.00 0.165387
\(433\) 17715.9 1.96622 0.983110 0.183014i \(-0.0585852\pi\)
0.983110 + 0.183014i \(0.0585852\pi\)
\(434\) − 1049.06i − 0.116029i
\(435\) − 1465.05i − 0.161480i
\(436\) − 4315.64i − 0.474041i
\(437\) 12041.1i 1.31808i
\(438\) 4820.46 0.525869
\(439\) −7163.47 −0.778801 −0.389401 0.921068i \(-0.627318\pi\)
−0.389401 + 0.921068i \(0.627318\pi\)
\(440\) − 405.566i − 0.0439423i
\(441\) 2685.01 0.289926
\(442\) 0 0
\(443\) −10169.2 −1.09064 −0.545321 0.838227i \(-0.683592\pi\)
−0.545321 + 0.838227i \(0.683592\pi\)
\(444\) − 8844.33i − 0.945346i
\(445\) −3265.20 −0.347833
\(446\) −11600.6 −1.23162
\(447\) 1141.35i 0.120770i
\(448\) − 4217.13i − 0.444733i
\(449\) − 17142.5i − 1.80179i −0.434037 0.900895i \(-0.642911\pi\)
0.434037 0.900895i \(-0.357089\pi\)
\(450\) − 4293.28i − 0.449750i
\(451\) 1808.73 0.188846
\(452\) 13924.8 1.44905
\(453\) 4552.36i 0.472160i
\(454\) 18629.7 1.92585
\(455\) 0 0
\(456\) 1251.70 0.128544
\(457\) − 14091.1i − 1.44235i −0.692750 0.721177i \(-0.743601\pi\)
0.692750 0.721177i \(-0.256399\pi\)
\(458\) −5381.76 −0.549068
\(459\) −779.181 −0.0792355
\(460\) − 3266.34i − 0.331074i
\(461\) 2922.22i 0.295231i 0.989045 + 0.147616i \(0.0471598\pi\)
−0.989045 + 0.147616i \(0.952840\pi\)
\(462\) 2666.01i 0.268472i
\(463\) − 2072.61i − 0.208040i −0.994575 0.104020i \(-0.966829\pi\)
0.994575 0.104020i \(-0.0331706\pi\)
\(464\) −8806.13 −0.881065
\(465\) 348.352 0.0347407
\(466\) − 13855.9i − 1.37739i
\(467\) 2664.19 0.263992 0.131996 0.991250i \(-0.457861\pi\)
0.131996 + 0.991250i \(0.457861\pi\)
\(468\) 0 0
\(469\) 382.114 0.0376213
\(470\) − 6514.01i − 0.639295i
\(471\) −4350.47 −0.425603
\(472\) −2706.91 −0.263974
\(473\) − 4124.08i − 0.400899i
\(474\) 11154.9i 1.08093i
\(475\) 11707.8i 1.13093i
\(476\) 1735.82i 0.167145i
\(477\) −6259.91 −0.600884
\(478\) 19531.8 1.86897
\(479\) − 5220.70i − 0.497995i −0.968504 0.248998i \(-0.919899\pi\)
0.968504 0.248998i \(-0.0801011\pi\)
\(480\) 2376.81 0.226013
\(481\) 0 0
\(482\) 19730.2 1.86449
\(483\) 2385.72i 0.224749i
\(484\) −2618.46 −0.245911
\(485\) 5351.28 0.501008
\(486\) 1001.91i 0.0935139i
\(487\) 12224.6i 1.13747i 0.822520 + 0.568737i \(0.192568\pi\)
−0.822520 + 0.568737i \(0.807432\pi\)
\(488\) 2891.55i 0.268226i
\(489\) − 7027.08i − 0.649848i
\(490\) 3751.77 0.345893
\(491\) −19653.2 −1.80639 −0.903195 0.429231i \(-0.858784\pi\)
−0.903195 + 0.429231i \(0.858784\pi\)
\(492\) − 1514.29i − 0.138759i
\(493\) 4620.59 0.422111
\(494\) 0 0
\(495\) −885.279 −0.0803845
\(496\) − 2093.88i − 0.189552i
\(497\) −2066.63 −0.186522
\(498\) 8498.90 0.764749
\(499\) 11713.6i 1.05084i 0.850842 + 0.525422i \(0.176093\pi\)
−0.850842 + 0.525422i \(0.823907\pi\)
\(500\) − 6607.26i − 0.590972i
\(501\) 120.219i 0.0107206i
\(502\) − 13498.5i − 1.20013i
\(503\) 13003.3 1.15266 0.576332 0.817216i \(-0.304483\pi\)
0.576332 + 0.817216i \(0.304483\pi\)
\(504\) 248.001 0.0219184
\(505\) 1954.60i 0.172235i
\(506\) 15822.1 1.39008
\(507\) 0 0
\(508\) −22456.4 −1.96130
\(509\) 5328.93i 0.464049i 0.972710 + 0.232024i \(0.0745349\pi\)
−0.972710 + 0.232024i \(0.925465\pi\)
\(510\) −1088.75 −0.0945308
\(511\) −2604.54 −0.225475
\(512\) − 16100.7i − 1.38976i
\(513\) − 2732.23i − 0.235148i
\(514\) − 26989.1i − 2.31603i
\(515\) 2114.16i 0.180895i
\(516\) −3452.72 −0.294569
\(517\) 16704.9 1.42105
\(518\) 9026.37i 0.765629i
\(519\) 5728.62 0.484506
\(520\) 0 0
\(521\) −11700.3 −0.983876 −0.491938 0.870630i \(-0.663711\pi\)
−0.491938 + 0.870630i \(0.663711\pi\)
\(522\) − 5941.41i − 0.498177i
\(523\) −4535.04 −0.379165 −0.189583 0.981865i \(-0.560714\pi\)
−0.189583 + 0.981865i \(0.560714\pi\)
\(524\) −393.814 −0.0328318
\(525\) 2319.70i 0.192838i
\(526\) − 363.664i − 0.0301454i
\(527\) 1098.66i 0.0908128i
\(528\) 5321.24i 0.438594i
\(529\) 1991.62 0.163690
\(530\) −8746.98 −0.716877
\(531\) 5908.71i 0.482893i
\(532\) −6086.73 −0.496040
\(533\) 0 0
\(534\) −13241.8 −1.07309
\(535\) − 1237.33i − 0.0999900i
\(536\) −235.739 −0.0189969
\(537\) −1529.88 −0.122940
\(538\) 18667.8i 1.49596i
\(539\) 9621.27i 0.768863i
\(540\) 741.164i 0.0590641i
\(541\) − 5184.89i − 0.412044i −0.978547 0.206022i \(-0.933948\pi\)
0.978547 0.206022i \(-0.0660519\pi\)
\(542\) 34311.7 2.71922
\(543\) −6410.64 −0.506643
\(544\) 7496.18i 0.590801i
\(545\) −1462.55 −0.114952
\(546\) 0 0
\(547\) 5609.12 0.438443 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(548\) 1855.75i 0.144660i
\(549\) 6311.74 0.490671
\(550\) 15384.2 1.19270
\(551\) 16202.3i 1.25271i
\(552\) − 1471.83i − 0.113487i
\(553\) − 6027.08i − 0.463468i
\(554\) − 11879.9i − 0.911066i
\(555\) −2997.30 −0.229241
\(556\) 900.000 0.0686484
\(557\) − 20150.5i − 1.53286i −0.642326 0.766432i \(-0.722030\pi\)
0.642326 0.766432i \(-0.277970\pi\)
\(558\) 1412.72 0.107178
\(559\) 0 0
\(560\) −1121.14 −0.0846011
\(561\) − 2792.06i − 0.210127i
\(562\) −11618.9 −0.872087
\(563\) −16292.2 −1.21960 −0.609800 0.792556i \(-0.708750\pi\)
−0.609800 + 0.792556i \(0.708750\pi\)
\(564\) − 13985.5i − 1.04414i
\(565\) − 4719.06i − 0.351385i
\(566\) 1090.98i 0.0810200i
\(567\) − 541.343i − 0.0400957i
\(568\) 1274.97 0.0941843
\(569\) −10460.5 −0.770700 −0.385350 0.922770i \(-0.625919\pi\)
−0.385350 + 0.922770i \(0.625919\pi\)
\(570\) − 3817.75i − 0.280541i
\(571\) −2225.96 −0.163141 −0.0815705 0.996668i \(-0.525994\pi\)
−0.0815705 + 0.996668i \(0.525994\pi\)
\(572\) 0 0
\(573\) −12173.2 −0.887511
\(574\) 1545.45i 0.112380i
\(575\) 13766.8 0.998461
\(576\) 5679.00 0.410807
\(577\) − 4686.23i − 0.338112i −0.985606 0.169056i \(-0.945928\pi\)
0.985606 0.169056i \(-0.0540718\pi\)
\(578\) 16823.0i 1.21063i
\(579\) − 2620.18i − 0.188067i
\(580\) − 4395.14i − 0.314652i
\(581\) −4592.03 −0.327899
\(582\) 21701.8 1.54565
\(583\) − 22431.3i − 1.59350i
\(584\) 1606.82 0.113854
\(585\) 0 0
\(586\) 17698.5 1.24764
\(587\) − 12090.6i − 0.850138i −0.905161 0.425069i \(-0.860250\pi\)
0.905161 0.425069i \(-0.139750\pi\)
\(588\) 8055.03 0.564938
\(589\) −3852.50 −0.269507
\(590\) 8256.25i 0.576109i
\(591\) 12441.7i 0.865964i
\(592\) 18016.2i 1.25078i
\(593\) 6135.97i 0.424914i 0.977170 + 0.212457i \(0.0681466\pi\)
−0.977170 + 0.212457i \(0.931853\pi\)
\(594\) −3590.19 −0.247992
\(595\) 588.261 0.0405317
\(596\) 3424.06i 0.235327i
\(597\) −7212.18 −0.494431
\(598\) 0 0
\(599\) −6198.80 −0.422831 −0.211416 0.977396i \(-0.567807\pi\)
−0.211416 + 0.977396i \(0.567807\pi\)
\(600\) − 1431.09i − 0.0973737i
\(601\) 18345.4 1.24513 0.622565 0.782568i \(-0.286091\pi\)
0.622565 + 0.782568i \(0.286091\pi\)
\(602\) 3523.79 0.238569
\(603\) 514.575i 0.0347514i
\(604\) 13657.1i 0.920030i
\(605\) 887.384i 0.0596319i
\(606\) 7926.75i 0.531357i
\(607\) 10388.1 0.694631 0.347315 0.937748i \(-0.387093\pi\)
0.347315 + 0.937748i \(0.387093\pi\)
\(608\) −26285.7 −1.75333
\(609\) 3210.19i 0.213602i
\(610\) 8819.40 0.585389
\(611\) 0 0
\(612\) −2337.54 −0.154395
\(613\) 804.480i 0.0530060i 0.999649 + 0.0265030i \(0.00843715\pi\)
−0.999649 + 0.0265030i \(0.991563\pi\)
\(614\) 28970.0 1.90413
\(615\) −513.184 −0.0336481
\(616\) 888.671i 0.0581259i
\(617\) − 15218.3i − 0.992973i −0.868044 0.496486i \(-0.834623\pi\)
0.868044 0.496486i \(-0.165377\pi\)
\(618\) 8573.83i 0.558074i
\(619\) − 11462.5i − 0.744291i −0.928174 0.372145i \(-0.878622\pi\)
0.928174 0.372145i \(-0.121378\pi\)
\(620\) 1045.05 0.0676942
\(621\) −3212.73 −0.207605
\(622\) − 4672.33i − 0.301195i
\(623\) 7154.66 0.460105
\(624\) 0 0
\(625\) 12223.0 0.782270
\(626\) 21794.5i 1.39151i
\(627\) 9790.49 0.623596
\(628\) −13051.4 −0.829311
\(629\) − 9453.14i − 0.599239i
\(630\) − 756.419i − 0.0478356i
\(631\) − 4468.68i − 0.281926i −0.990015 0.140963i \(-0.954980\pi\)
0.990015 0.140963i \(-0.0450199\pi\)
\(632\) 3718.30i 0.234028i
\(633\) −11606.6 −0.728783
\(634\) 19717.3 1.23514
\(635\) 7610.37i 0.475603i
\(636\) −18779.7 −1.17086
\(637\) 0 0
\(638\) 21290.0 1.32113
\(639\) − 2783.04i − 0.172293i
\(640\) 1597.12 0.0986430
\(641\) −6142.36 −0.378484 −0.189242 0.981930i \(-0.560603\pi\)
−0.189242 + 0.981930i \(0.560603\pi\)
\(642\) − 5017.93i − 0.308477i
\(643\) − 20738.2i − 1.27190i −0.771729 0.635951i \(-0.780608\pi\)
0.771729 0.635951i \(-0.219392\pi\)
\(644\) 7157.15i 0.437937i
\(645\) 1170.11i 0.0714312i
\(646\) 12040.7 0.733339
\(647\) −852.757 −0.0518166 −0.0259083 0.999664i \(-0.508248\pi\)
−0.0259083 + 0.999664i \(0.508248\pi\)
\(648\) 333.972i 0.0202464i
\(649\) −21172.8 −1.28060
\(650\) 0 0
\(651\) −763.303 −0.0459542
\(652\) − 21081.3i − 1.26627i
\(653\) −7345.75 −0.440217 −0.220108 0.975475i \(-0.570641\pi\)
−0.220108 + 0.975475i \(0.570641\pi\)
\(654\) −5931.28 −0.354635
\(655\) 133.462i 0.00796151i
\(656\) 3084.65i 0.183591i
\(657\) − 3507.40i − 0.208275i
\(658\) 14273.4i 0.845645i
\(659\) 12540.7 0.741297 0.370648 0.928773i \(-0.379136\pi\)
0.370648 + 0.928773i \(0.379136\pi\)
\(660\) −2655.84 −0.156634
\(661\) − 2242.95i − 0.131983i −0.997820 0.0659915i \(-0.978979\pi\)
0.997820 0.0659915i \(-0.0210210\pi\)
\(662\) −35747.3 −2.09873
\(663\) 0 0
\(664\) 2832.97 0.165573
\(665\) 2062.76i 0.120287i
\(666\) −12155.4 −0.707224
\(667\) 19051.7 1.10597
\(668\) 360.658i 0.0208896i
\(669\) 8440.66i 0.487795i
\(670\) 719.016i 0.0414597i
\(671\) 22617.0i 1.30122i
\(672\) −5208.03 −0.298964
\(673\) 4776.46 0.273579 0.136790 0.990600i \(-0.456322\pi\)
0.136790 + 0.990600i \(0.456322\pi\)
\(674\) 35156.0i 2.00914i
\(675\) −3123.82 −0.178127
\(676\) 0 0
\(677\) −7933.57 −0.450387 −0.225193 0.974314i \(-0.572301\pi\)
−0.225193 + 0.974314i \(0.572301\pi\)
\(678\) − 19137.9i − 1.08405i
\(679\) −11725.6 −0.662723
\(680\) −362.917 −0.0204665
\(681\) − 13555.1i − 0.762750i
\(682\) 5062.23i 0.284227i
\(683\) 23573.8i 1.32068i 0.750967 + 0.660340i \(0.229588\pi\)
−0.750967 + 0.660340i \(0.770412\pi\)
\(684\) − 8196.70i − 0.458200i
\(685\) 628.904 0.0350791
\(686\) −17672.4 −0.983581
\(687\) 3915.81i 0.217463i
\(688\) 7033.32 0.389743
\(689\) 0 0
\(690\) −4489.15 −0.247680
\(691\) 12543.8i 0.690575i 0.938497 + 0.345288i \(0.112219\pi\)
−0.938497 + 0.345288i \(0.887781\pi\)
\(692\) 17185.9 0.944087
\(693\) 1939.81 0.106331
\(694\) 51874.8i 2.83738i
\(695\) − 305.006i − 0.0166468i
\(696\) − 1980.47i − 0.107858i
\(697\) − 1618.52i − 0.0879568i
\(698\) −36947.9 −2.00358
\(699\) −10081.6 −0.545526
\(700\) 6959.09i 0.375755i
\(701\) 581.786 0.0313463 0.0156731 0.999877i \(-0.495011\pi\)
0.0156731 + 0.999877i \(0.495011\pi\)
\(702\) 0 0
\(703\) 33147.9 1.77837
\(704\) 20349.7i 1.08943i
\(705\) −4739.64 −0.253199
\(706\) 22091.6 1.17766
\(707\) − 4282.89i − 0.227828i
\(708\) 17726.1i 0.940944i
\(709\) − 20742.0i − 1.09871i −0.835590 0.549353i \(-0.814874\pi\)
0.835590 0.549353i \(-0.185126\pi\)
\(710\) − 3888.74i − 0.205552i
\(711\) 8116.38 0.428113
\(712\) −4413.94 −0.232331
\(713\) 4530.01i 0.237938i
\(714\) 2385.66 0.125043
\(715\) 0 0
\(716\) −4589.63 −0.239556
\(717\) − 14211.5i − 0.740221i
\(718\) −11154.6 −0.579788
\(719\) −25350.2 −1.31489 −0.657443 0.753504i \(-0.728362\pi\)
−0.657443 + 0.753504i \(0.728362\pi\)
\(720\) − 1509.78i − 0.0781474i
\(721\) − 4632.51i − 0.239284i
\(722\) 13941.0i 0.718602i
\(723\) − 14355.8i − 0.738449i
\(724\) −19231.9 −0.987223
\(725\) 18524.4 0.948938
\(726\) 3598.73i 0.183969i
\(727\) −33428.2 −1.70534 −0.852672 0.522447i \(-0.825019\pi\)
−0.852672 + 0.522447i \(0.825019\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 4900.90i − 0.248480i
\(731\) −3690.39 −0.186722
\(732\) 18935.2 0.956101
\(733\) 3842.67i 0.193632i 0.995302 + 0.0968160i \(0.0308658\pi\)
−0.995302 + 0.0968160i \(0.969134\pi\)
\(734\) − 43184.4i − 2.17162i
\(735\) − 2729.81i − 0.136994i
\(736\) 30908.3i 1.54796i
\(737\) −1843.89 −0.0921582
\(738\) −2081.19 −0.103807
\(739\) 29029.6i 1.44502i 0.691359 + 0.722511i \(0.257012\pi\)
−0.691359 + 0.722511i \(0.742988\pi\)
\(740\) −8991.91 −0.446688
\(741\) 0 0
\(742\) 19166.3 0.948269
\(743\) 34996.7i 1.72800i 0.503492 + 0.864000i \(0.332048\pi\)
−0.503492 + 0.864000i \(0.667952\pi\)
\(744\) 470.906 0.0232046
\(745\) 1160.40 0.0570653
\(746\) 52623.2i 2.58267i
\(747\) − 6183.86i − 0.302886i
\(748\) − 8376.19i − 0.409444i
\(749\) 2711.23i 0.132265i
\(750\) −9080.82 −0.442113
\(751\) −10454.1 −0.507957 −0.253979 0.967210i \(-0.581739\pi\)
−0.253979 + 0.967210i \(0.581739\pi\)
\(752\) 28489.1i 1.38150i
\(753\) −9821.58 −0.475323
\(754\) 0 0
\(755\) 4628.32 0.223102
\(756\) − 1624.03i − 0.0781287i
\(757\) −28130.4 −1.35062 −0.675308 0.737536i \(-0.735989\pi\)
−0.675308 + 0.737536i \(0.735989\pi\)
\(758\) 9557.45 0.457971
\(759\) − 11512.3i − 0.550552i
\(760\) − 1272.58i − 0.0607388i
\(761\) 21087.0i 1.00447i 0.864731 + 0.502236i \(0.167489\pi\)
−0.864731 + 0.502236i \(0.832511\pi\)
\(762\) 30863.4i 1.46727i
\(763\) 3204.72 0.152056
\(764\) −36519.7 −1.72937
\(765\) 792.183i 0.0374398i
\(766\) −8177.54 −0.385727
\(767\) 0 0
\(768\) −8667.00 −0.407218
\(769\) − 19527.9i − 0.915728i −0.889022 0.457864i \(-0.848615\pi\)
0.889022 0.457864i \(-0.151385\pi\)
\(770\) 2710.50 0.126857
\(771\) −19637.4 −0.917283
\(772\) − 7860.55i − 0.366460i
\(773\) 29352.0i 1.36574i 0.730540 + 0.682870i \(0.239269\pi\)
−0.730540 + 0.682870i \(0.760731\pi\)
\(774\) 4745.31i 0.220370i
\(775\) 4404.64i 0.204154i
\(776\) 7233.92 0.334643
\(777\) 6567.65 0.303234
\(778\) − 13377.5i − 0.616459i
\(779\) 5675.42 0.261031
\(780\) 0 0
\(781\) 9972.54 0.456908
\(782\) − 14158.3i − 0.647440i
\(783\) −4323.01 −0.197307
\(784\) −16408.4 −0.747467
\(785\) 4423.06i 0.201103i
\(786\) 541.246i 0.0245618i
\(787\) 4463.12i 0.202151i 0.994879 + 0.101076i \(0.0322284\pi\)
−0.994879 + 0.101076i \(0.967772\pi\)
\(788\) 37325.2i 1.68738i
\(789\) −264.604 −0.0119394
\(790\) 11341.0 0.510754
\(791\) 10340.3i 0.464804i
\(792\) −1196.73 −0.0536919
\(793\) 0 0
\(794\) 15501.7 0.692866
\(795\) 6364.37i 0.283926i
\(796\) −21636.6 −0.963426
\(797\) 34785.5 1.54600 0.773002 0.634404i \(-0.218754\pi\)
0.773002 + 0.634404i \(0.218754\pi\)
\(798\) 8365.40i 0.371093i
\(799\) − 14948.2i − 0.661866i
\(800\) 30053.0i 1.32817i
\(801\) 9634.84i 0.425007i
\(802\) −8236.09 −0.362627
\(803\) 12568.2 0.552330
\(804\) 1543.72i 0.0677152i
\(805\) 2425.53 0.106197
\(806\) 0 0
\(807\) 13582.8 0.592487
\(808\) 2642.25i 0.115042i
\(809\) −10620.0 −0.461530 −0.230765 0.973010i \(-0.574123\pi\)
−0.230765 + 0.973010i \(0.574123\pi\)
\(810\) 1018.63 0.0441866
\(811\) − 5497.87i − 0.238047i −0.992891 0.119024i \(-0.962024\pi\)
0.992891 0.119024i \(-0.0379764\pi\)
\(812\) 9630.57i 0.416215i
\(813\) − 24965.5i − 1.07697i
\(814\) − 43556.7i − 1.87551i
\(815\) −7144.34 −0.307062
\(816\) 4761.66 0.204279
\(817\) − 12940.5i − 0.554139i
\(818\) 21420.5 0.915587
\(819\) 0 0
\(820\) −1539.55 −0.0655653
\(821\) − 21305.3i − 0.905678i −0.891592 0.452839i \(-0.850411\pi\)
0.891592 0.452839i \(-0.149589\pi\)
\(822\) 2550.48 0.108222
\(823\) −17342.6 −0.734537 −0.367268 0.930115i \(-0.619707\pi\)
−0.367268 + 0.930115i \(0.619707\pi\)
\(824\) 2857.94i 0.120827i
\(825\) − 11193.7i − 0.472381i
\(826\) − 18091.0i − 0.762064i
\(827\) 5129.96i 0.215703i 0.994167 + 0.107851i \(0.0343971\pi\)
−0.994167 + 0.107851i \(0.965603\pi\)
\(828\) −9638.19 −0.404529
\(829\) 8471.81 0.354931 0.177466 0.984127i \(-0.443210\pi\)
0.177466 + 0.984127i \(0.443210\pi\)
\(830\) − 8640.72i − 0.361354i
\(831\) −8643.93 −0.360836
\(832\) 0 0
\(833\) 8609.50 0.358105
\(834\) − 1236.93i − 0.0513566i
\(835\) 122.225 0.00506561
\(836\) 29371.5 1.21511
\(837\) − 1027.90i − 0.0424486i
\(838\) 28128.3i 1.15952i
\(839\) − 19155.0i − 0.788207i −0.919066 0.394103i \(-0.871055\pi\)
0.919066 0.394103i \(-0.128945\pi\)
\(840\) − 252.140i − 0.0103567i
\(841\) 1246.67 0.0511160
\(842\) −31078.7 −1.27202
\(843\) 8453.98i 0.345398i
\(844\) −34819.7 −1.42007
\(845\) 0 0
\(846\) −19221.3 −0.781136
\(847\) − 1944.42i − 0.0788797i
\(848\) 38255.0 1.54915
\(849\) 793.804 0.0320887
\(850\) − 13766.4i − 0.555512i
\(851\) − 38977.3i − 1.57006i
\(852\) − 8349.11i − 0.335723i
\(853\) − 18075.1i − 0.725532i −0.931880 0.362766i \(-0.881832\pi\)
0.931880 0.362766i \(-0.118168\pi\)
\(854\) −19324.9 −0.774339
\(855\) −2777.82 −0.111111
\(856\) − 1672.64i − 0.0667871i
\(857\) 21054.6 0.839219 0.419609 0.907705i \(-0.362167\pi\)
0.419609 + 0.907705i \(0.362167\pi\)
\(858\) 0 0
\(859\) 920.322 0.0365553 0.0182776 0.999833i \(-0.494182\pi\)
0.0182776 + 0.999833i \(0.494182\pi\)
\(860\) 3510.33i 0.139188i
\(861\) 1124.48 0.0445090
\(862\) 55267.0 2.18376
\(863\) 19427.5i 0.766304i 0.923685 + 0.383152i \(0.125161\pi\)
−0.923685 + 0.383152i \(0.874839\pi\)
\(864\) − 7013.40i − 0.276158i
\(865\) − 5824.21i − 0.228935i
\(866\) − 73044.7i − 2.86623i
\(867\) 12240.5 0.479482
\(868\) −2289.91 −0.0895444
\(869\) 29083.7i 1.13532i
\(870\) −6040.55 −0.235395
\(871\) 0 0
\(872\) −1977.09 −0.0767808
\(873\) − 15790.3i − 0.612168i
\(874\) 49646.5 1.92142
\(875\) 4906.44 0.189563
\(876\) − 10522.2i − 0.405836i
\(877\) − 14872.2i − 0.572632i −0.958135 0.286316i \(-0.907569\pi\)
0.958135 0.286316i \(-0.0924306\pi\)
\(878\) 29535.7i 1.13529i
\(879\) − 12877.6i − 0.494141i
\(880\) 5410.04 0.207241
\(881\) 12940.6 0.494870 0.247435 0.968905i \(-0.420412\pi\)
0.247435 + 0.968905i \(0.420412\pi\)
\(882\) − 11070.6i − 0.422637i
\(883\) 25585.5 0.975108 0.487554 0.873093i \(-0.337889\pi\)
0.487554 + 0.873093i \(0.337889\pi\)
\(884\) 0 0
\(885\) 6007.30 0.228173
\(886\) 41928.8i 1.58987i
\(887\) 3716.46 0.140684 0.0703418 0.997523i \(-0.477591\pi\)
0.0703418 + 0.997523i \(0.477591\pi\)
\(888\) −4051.79 −0.153118
\(889\) − 16675.7i − 0.629118i
\(890\) 13462.8i 0.507049i
\(891\) 2612.25i 0.0982195i
\(892\) 25322.0i 0.950496i
\(893\) 52416.7 1.96423
\(894\) 4705.92 0.176051
\(895\) 1555.40i 0.0580910i
\(896\) −3499.58 −0.130483
\(897\) 0 0
\(898\) −70680.3 −2.62654
\(899\) 6095.52i 0.226137i
\(900\) −9371.47 −0.347091
\(901\) −20072.5 −0.742187
\(902\) − 7457.57i − 0.275288i
\(903\) − 2563.93i − 0.0944876i
\(904\) − 6379.29i − 0.234703i
\(905\) 6517.61i 0.239395i
\(906\) 18769.8 0.688285
\(907\) 12960.4 0.474469 0.237235 0.971452i \(-0.423759\pi\)
0.237235 + 0.971452i \(0.423759\pi\)
\(908\) − 40665.3i − 1.48626i
\(909\) 5767.56 0.210449
\(910\) 0 0
\(911\) −36607.1 −1.33134 −0.665668 0.746248i \(-0.731853\pi\)
−0.665668 + 0.746248i \(0.731853\pi\)
\(912\) 16697.0i 0.606242i
\(913\) 22158.8 0.803230
\(914\) −58099.3 −2.10258
\(915\) − 6417.06i − 0.231849i
\(916\) 11747.4i 0.423740i
\(917\) − 292.440i − 0.0105313i
\(918\) 3212.65i 0.115505i
\(919\) 20356.3 0.730676 0.365338 0.930875i \(-0.380953\pi\)
0.365338 + 0.930875i \(0.380953\pi\)
\(920\) −1496.38 −0.0536243
\(921\) − 21078.8i − 0.754147i
\(922\) 12048.6 0.430370
\(923\) 0 0
\(924\) 5819.43 0.207192
\(925\) − 37898.7i − 1.34714i
\(926\) −8545.61 −0.303268
\(927\) 6238.38 0.221030
\(928\) 41589.8i 1.47118i
\(929\) 45069.7i 1.59170i 0.605495 + 0.795849i \(0.292975\pi\)
−0.605495 + 0.795849i \(0.707025\pi\)
\(930\) − 1436.29i − 0.0506428i
\(931\) 30189.6i 1.06275i
\(932\) −30244.9 −1.06299
\(933\) −3399.62 −0.119291
\(934\) − 10984.7i − 0.384831i
\(935\) −2838.65 −0.0992876
\(936\) 0 0
\(937\) −6771.10 −0.236075 −0.118037 0.993009i \(-0.537660\pi\)
−0.118037 + 0.993009i \(0.537660\pi\)
\(938\) − 1575.50i − 0.0548420i
\(939\) 15857.8 0.551119
\(940\) −14218.9 −0.493372
\(941\) − 36690.7i − 1.27108i −0.772070 0.635538i \(-0.780778\pi\)
0.772070 0.635538i \(-0.219222\pi\)
\(942\) 17937.4i 0.620417i
\(943\) − 6673.51i − 0.230455i
\(944\) − 36108.8i − 1.24496i
\(945\) −550.376 −0.0189457
\(946\) −17004.0 −0.584406
\(947\) − 50861.2i − 1.74527i −0.488375 0.872634i \(-0.662410\pi\)
0.488375 0.872634i \(-0.337590\pi\)
\(948\) 24349.1 0.834202
\(949\) 0 0
\(950\) 48272.6 1.64860
\(951\) − 14346.5i − 0.489187i
\(952\) 795.219 0.0270727
\(953\) −11855.6 −0.402980 −0.201490 0.979491i \(-0.564578\pi\)
−0.201490 + 0.979491i \(0.564578\pi\)
\(954\) 25810.3i 0.875931i
\(955\) 12376.4i 0.419361i
\(956\) − 42634.5i − 1.44236i
\(957\) − 15490.8i − 0.523245i
\(958\) −21525.5 −0.725947
\(959\) −1378.04 −0.0464019
\(960\) − 5773.76i − 0.194112i
\(961\) 28341.6 0.951349
\(962\) 0 0
\(963\) −3651.08 −0.122175
\(964\) − 43067.5i − 1.43891i
\(965\) −2663.90 −0.0888643
\(966\) 9836.56 0.327625
\(967\) − 40661.7i − 1.35221i −0.736803 0.676107i \(-0.763666\pi\)
0.736803 0.676107i \(-0.236334\pi\)
\(968\) 1199.58i 0.0398304i
\(969\) − 8760.93i − 0.290445i
\(970\) − 22063.9i − 0.730339i
\(971\) 57318.3 1.89437 0.947184 0.320690i \(-0.103915\pi\)
0.947184 + 0.320690i \(0.103915\pi\)
\(972\) 2187.00 0.0721688
\(973\) 668.324i 0.0220200i
\(974\) 50403.3 1.65814
\(975\) 0 0
\(976\) −38571.7 −1.26501
\(977\) 3026.73i 0.0991134i 0.998771 + 0.0495567i \(0.0157808\pi\)
−0.998771 + 0.0495567i \(0.984219\pi\)
\(978\) −28973.4 −0.947308
\(979\) −34524.8 −1.12709
\(980\) − 8189.43i − 0.266941i
\(981\) 4315.64i 0.140457i
\(982\) 81032.3i 2.63324i
\(983\) 33942.5i 1.10132i 0.834730 + 0.550659i \(0.185624\pi\)
−0.834730 + 0.550659i \(0.814376\pi\)
\(984\) −693.729 −0.0224749
\(985\) 12649.3 0.409179
\(986\) − 19051.2i − 0.615327i
\(987\) 10385.4 0.334925
\(988\) 0 0
\(989\) −15216.3 −0.489231
\(990\) 3650.10i 0.117179i
\(991\) 21637.5 0.693580 0.346790 0.937943i \(-0.387272\pi\)
0.346790 + 0.937943i \(0.387272\pi\)
\(992\) −9889.02 −0.316509
\(993\) 26009.9i 0.831219i
\(994\) 8520.95i 0.271900i
\(995\) 7332.53i 0.233625i
\(996\) − 18551.6i − 0.590190i
\(997\) 19624.6 0.623388 0.311694 0.950182i \(-0.399104\pi\)
0.311694 + 0.950182i \(0.399104\pi\)
\(998\) 48296.3 1.53186
\(999\) 8844.33i 0.280102i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.4.b.e.337.1 4
13.3 even 3 39.4.j.b.4.2 4
13.4 even 6 39.4.j.b.10.2 yes 4
13.5 odd 4 507.4.a.k.1.1 4
13.8 odd 4 507.4.a.k.1.4 4
13.12 even 2 inner 507.4.b.e.337.4 4
39.5 even 4 1521.4.a.z.1.4 4
39.8 even 4 1521.4.a.z.1.1 4
39.17 odd 6 117.4.q.d.10.1 4
39.29 odd 6 117.4.q.d.82.1 4
52.3 odd 6 624.4.bv.c.433.2 4
52.43 odd 6 624.4.bv.c.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.2 4 13.3 even 3
39.4.j.b.10.2 yes 4 13.4 even 6
117.4.q.d.10.1 4 39.17 odd 6
117.4.q.d.82.1 4 39.29 odd 6
507.4.a.k.1.1 4 13.5 odd 4
507.4.a.k.1.4 4 13.8 odd 4
507.4.b.e.337.1 4 1.1 even 1 trivial
507.4.b.e.337.4 4 13.12 even 2 inner
624.4.bv.c.49.1 4 52.43 odd 6
624.4.bv.c.433.2 4 52.3 odd 6
1521.4.a.z.1.1 4 39.8 even 4
1521.4.a.z.1.4 4 39.5 even 4